1. Technical Field
This invention relates to business models for determining returns attainable on a set of options traded on options markets based upon a volatility factor.
2. Description of the Prior Art
An option is a right to buy or sell a specific amount or value of a particular underlying interest at a fixed price through exercising the option before its stated expiration date. In actuality, an option is a contract between two parties, the buyer of the option and the seller of the option. The buyer of an option is also known as its owner or holder, and the seller of an option is also known as its writer or assigned writer if the seller has been assigned an exercise right or obligation. An option which provides the right to buy the underlying interest is called a “call” option, and an option which provides the right to sell the underlying interest is called a “put” option. The buying and selling of call and put options are independent of one another and, irrespective of the type and classification of the option, options contracts give the buyer/owner the right, but not the obligation, to buy or sell an underlying interest.
There are two kinds of options. Physical delivery options and cash-settled options. Physical delivery options provide the option owner with the right to physical delivery of the underlying interest if the option is a call option or the right to make physical delivery of the underlying interest if the option is a put option. Cash-settled options provide the option owner with the right to receive a cash payment based on the difference between the underlying value of the underlying interest at the time of option exercise and the fixed exercise price of the option. A cash-settled call option allows the option owner to receive a payment if the value of the underlying interest at exercise exceeds the fixed exercise price, and a cash-settled put option allows the option owner to receive a payment if the exercise price exceeds the value of the underlying interest. Irrespective of the type and classification of the option, options contracts give the buyer/owner the right, but not the obligation, to buy or sell an underlying security at a fixed price for a specific period of time.
The most commonly known and traded options are options traded on various options exchanges in the United States and throughout the world. Examples of such exchanges are the New York Stock Exchange, American Stock Exchange, Chicago Board Options Exchange, Pacific Stock Exchange, Philadelphia Stock Exchange, and the European Options Exchange in Amsterdam. However, many additional options markets may exist. As a result of the existence of options markets, options contracts have become relatively uniform in standardized terms which allows for easy trading of options on the options markets. However, each U.S. options market publishes its own specifications for options traded on that market which sets out the standardized terms of the options traded on that options market. Options with the same standardized terms are identical and comprise an options “series.” Examples of the standardized terms are such terms as the form and amount of the underlying interest, the expiration date, the exercise price, classification of the option as a call or put and as a physical delivery option or a cash-settled option, the specific terms of any cash-settled option, the style of option, and if there is an automatic exercise provision or adjustment provision to the option. The use of standardized terms increase the likelihood that the options markets will function as a secondary market for the purchase and sale of options contracts. This provides holders and writers of options with a means to close out their option positions by offsetting sales and purchases. By selling an option of the same series as the one bought, an options holder may closeout his position in that option. By buying an option of the same series as the one sold, an option writer may closeout his position in that option as well. Accordingly, standardized terms and the existence of options series increases the ability of investors to trade options and create a market for the buying and selling of options at prices determined by market factors.
Individual options markets determine the availability of options for selected underlying interests. There are four common types of underlying interest for which options are available: equity securities, futures and stock indexes, government debt securities, and foreign currencies. However, one should keep in mind that these are not the only types of underlying interests which may exist.
The writer, or assigned writer, of an option is obligated to perform according to the terms of the options contract. The contract may have terms for selling the underlying interest at the contracted price for a writer of a call option, or purchasing the underlying interest for a writer of a put option. However, these terms will only impact the writer if the options contract is exercised by the holder. The price that a holder of an option pays is called its premium. The premium is a non-refundable payment for the rights contained in the option. The potential loss to the holder of an option can be no greater than the initial premium paid for the option. As such, a holder controls the amount of the assumed risk. Conversely, the seller of an option assumes the risk of being assigned if the contract is exercised. In such a case, depending upon whether or not the option was “covered” (i.e. the writer owned the underlying interest or is short the underlying interest) or uncovered or naked (i.e. the writer does not have a position in the underlying interest), the writer's risk may be substantial. All options of a series expire on a certain date, known as the expiration date. The option contract may be exercised at any time between the date of purchase of the contract and the expiration date of the contract. Accordingly, an option is a contract providing a holder with the right to purchase or sell an underlying interest without an obligation to actually purchase or sell the underlying interest.
Both call options and put options are known as derivative securities. Their value is derived from the value of an underlying interest. The most common type of underlying interest is the equity security. Equity securities include common stock as well as many other forms of equity such as limited partnership interests. Options on equity securities are commonly known as “stock options” since the majority of such options are on an underlying interest of stock in a corporation. Each physical delivery option contract will have a “unit of trading” or “contract size” which will fix the amount of the underlying interest to be delivered upon exercise. Cash-settled options have a contract size determined by a multilayer which determines the aggregate value of each point of the difference between the exercise price of the option and the settlement value of the underlying interest. Options will have a fixed exercise price, also known as the “strike price,” at which price the holder has the right to either buy or sell the underlying stock if the option is a physical delivery option or at which price the holder had the right to receive a cash settlement amount if the option is a cash-settlement option.
Writers and holders of options contracts take positions in the options. When a writer sells an option or a holder purchases an option they are said to have taken an “open” position in the option. A writer or holder is said to have “closed” their position when, prior to expiration, the holder has made an offsetting sale of an identical option (i.e. the same series) or a writer has made an offsetting purchase of an identical option. Two other common positions in options trading are spreads and straddles which are two types of combination positions. Combination positions are positions in more than one option at the same time which are not the same series and may be also used as a “hedge” or “spread” or “straddle”. A hedge position is when the investor owns or is short the underlying interest and has an opposing position in options on that underlying interest; that is, long the underlying interest and short call options or long put options on that interest, or short the underlying interest and long calls or short puts on that interest. A spread position is when the investor is both the holder and writer of the same type of option in the same underlying interest with the options having different exercise prices and/or expiration dates. A straddle position is when the investor either holds or writes both a put and call option on the same underlying interest, with the same exercise price and expiration date.
Various methods are known for calculating a fair price for options. One common method for calculating a fair price for options is known as the “Black Scholes” model. A common saying in the investment community is “Buy low and sell high”. This logic is applied equally to the options trading markets, as options traders commonly attempt to purchase options for less than the fair value calculated by the investor's Black Scholes model and to sell options for more than the fair value calculated by their Black Scholes model.
One factor in determining a fair price for an option is volatility of the underlying interest, which is usually computed as a measure of changes in price expressed in percentage terms. The volatility that affects the value of the option and that is used in options models like the Black Scholes model is the volatility of the underlying between the time of the valuation and the expiration of the option. Mathematically, the volatility of the underlying interest over a specific period is usually measured as the annualized standard deviation of daily returns during that specific period. Accordingly, the future volatility is an annualized standard deviation of daily returns during some future period. However, future volatility by its nature cannot be known but only estimated. Various methods of estimating the future volatility are used. The most common is to use the volatility of the underlying interest over a recent time period, and perhaps extrapolating it using a trend analysis. Options prices are themselves used to estimate volatility, by using the “implied volatility”, which is that volatility value that “explains” the current market price of an option, meaning that volatility value which, if used in the option model such as Black Scholes, makes the theoretical fair value equal to the market price of the option. Different options series on the same underlying may, and usually do, yield different implied volatilities, so that difference options express different, and therefore inconsistent, market “opinions” about the future volatility of the underlying. Implied volatility of an options series is often used as a measure of the degree of pricing extremity of the options series, and the implied volatilities of different options are compared by investors and traders to determine which series on an underlying are the best “buys” or “writes”. Further, the implied volatilities of options on different underlyings are often compared with one another and with the estimates of future volatilities of the underlyings, in order to determine which of the underlyings offer the best opportunities for hedging or spreading. In determining a fair price for an option, it is common to evaluate the relationship between implied volatility and estimated future volatility. Volatility calculations used in options pricing formulas are only estimates of future volatility, but they are critical in determining the value of an option.
The Black Scholes model also provides a conventional method of calculating an implied interest rate. The Black Scholes method of options valuation has two significant suppositions: 1) a riskless hedge could be constructed between an underlying interest and a variable quantity of call options on the underlying interest, with the variable quantity selected continuously so that the behavior of the call option positions imitate the reverse underlying interest position and produce a riskless return, and 2) the rate of return on money invested in the riskless hedge should be equal to the rate of return on any other riskless investment. The Black Scholes equation is DNRR r, where DNRR is the “delta neutral rate of return” on a continuously adjusted hedge and r is the riskless rate of return, often taken to be the T-bill rate for the period of time until expiration of the option. In the case of a call option on a stock that pays no dividends, this equation leads to a formula that allows traders to derive the value of the call option given its strike price and expiration date and the volatility of the underlying between the time of the valuation and the expiration of the call. However, this mathematical formula has two uncertain factors, the future volatility of the underlying interest's price until expiration and the riskless interest rate. The volatility of the underlying interest's price was assumed by Black and Scholes to be known and constant. However, in actuality the future volatility of the underlying interest's price is unknown and difficult to ascertain with precision. In practice, most traders estimate the future volatility by looking at the recent past. If this practice is followed, often options are not priced by the market according to the Black Scholes model.
The only prior art calculation that attempts to measure the general implied volatility level of a collection of calls and puts on a given underlying is the Volatility Index (VIX) which was proposed by Robert Whaley of Duke University in a paper entitled “Derivative on Market Volatility: Hedging Tools Long Overdue” published in the Journal of Derivatives 1 (Fall 1993), 71-84. The broadcasting of VIX was introduced by the Chicago Board of Options Exchange in 1993 on a real time basis. VIX provides a simple measure of the general implied volatility level for the general class of options, both puts and calls, on the OEX index. VIX is created from eight options that are near-the-money and within thirty days of expiration. However, VIX has the following limitations: (1) VIX applies only to options on the OEX index, (2) it considers both calls and puts together and does not distinguish call levels from put levels, (3) it is not a uniquely defined number but is rather a function of the riskless interest rate assumed, (4) it uses a fixed interest rate and does not consider the interest rate to be a measure of return on delta neutral hedging, (5) it cannot be used to derive a unique and unequivocal estimate of future volatility for an arbitrary given time period, and (6) it cannot be used to derive a unique and unequivocal estimate of future volatility for an arbitrary expiration of the options on the underlying. Accordingly, there is a need to have a good estimate of future volatility of the underlying based upon the market prices of the options on the underlying, and the usual inconsistency of implied volatility estimates provided by the different options series on an underlying makes it highly desirable to have a method of estimating volatility from market prices that is consistent with all market prices of options on an underlying interest.